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We have $C_N=M_N(\\mathbb T)\\cap\\sqrt{N}U_N$, and following Tadej and \\.Zyczkowski we investigate here the computation of the enveloping tangent space $\\widetilde{T}_HC_N=T_HM_N(\\mathbb T)\\cap T_H\\sqrt{N}U_N$, and notably of its dimension $d(H)=\\dim(\\widetilde{T}_HC_N)$, called undephased defect of $H$. Our main result is an explicit formula for the defect of the Fourier matrix $F_G$ associated to an arbitrary finite abelian group $G=\\mathbb Z_{N_1}\\times...\\times\\mathbb Z_{N_r}$. 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